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was one of the first to detect the gaps in his predecessors*
reasoning, and he resolved to devote a fair share of his lifework
to caulking the cracks and making the reasoning watertight.
One of Ms classics in this direction is the first proof of the general
binomial theorem, special cases of which had been stated by
Newton and Euler. It is not easy to give a sound proof in the
general case, so perhaps it is not astonishing to find alleged
proofs still displayed in the schoolbooks as if Abel had never
lived. This proof, however, was only a detail in Abel's vaster
programme of cleaning up the theory and application of infinite
Abel's father died in 1820 at the age of forty-eight. At the
time Abel was eighteen. The care of his mother and six children
fell on his shoulders. Confident of himself Abel assumed his
sudden responsibilities cheerfully. Abel was a genial and opti-
mistic soul. With no more than strict justice he foresaw himself
as an honoured and moderately prosperous mathematician in a
university chair. Then he could provide for the lot of them in
reasonable security. In the meantime he took private pupils
and did what he could. In passing it may be noted that Abel
was a very successful teacher. Had he been footloose poverty
would never have bothered him. He could have earned enough
for his own modest needs, somehow or other, at any time. But
with seven on his back he had no chance* He never complained,
but took it all in his stride as part of the day's work and kept at
his mathematical researches in every spare moment.
Convinced that he had one of the greatest mathematicians of
all time on his hands, Holmboe did what he could by getting
subsidies for the young man and digging down generously into
his own none too deep pocket. But the country was poor to the
point of starvation and not nearly enough could be done. In
those days of privation and incessant work Abel immortalized
himself and sowed the seeds of the disease which was to kill
him before he had half done his work.
Abel's first ambitious venture was an attack on the general
equation of the fifth degree (the *quintie'). All his great pre-
decessors in algebra had exhausted their efforts to produce a
solution, without success. We can easily imagine Abel's exulta-